Snell's Law Calculator
Solve n₁ sin θ₁ = n₂ sin θ₂ in one step. Enter the refractive index of each medium (air 1.000, water 1.333, glass 1.52, diamond 2.417...) and the incidence angle in degrees from the normal — you get the refraction angle, the critical angle for total internal reflection, and how fast light moves in the second medium.
Example: with Refractive index n₁ (incident medium) 1 · Refractive index n₂ (second medium) 1.333 · Angle of incidence θ₁ (degrees from normal) 45 → Angle of refraction θ₂: 32.04° from the normal.
- Critical angle (n₁ → n₂)None — light entering a denser medium always refracts
- Light speed in medium 2224,900,569 m/s (75.0% of c)
Computed by the calculator below using its default values. Change any input to see your own numbers.
n₁ sin θ₁ = n₂ sin θ₂ (Snell's law, 1621). Angles are measured from the normal — the line perpendicular to the surface — not from the surface itself.
Why light bends at a boundary
Light travels at c/n inside a material with refractive index n. When a wavefront crosses into a slower medium at an angle, the edge that arrives first slows down first, swinging the whole front toward the normal — exactly like a car drifting onto a sandy shoulder pulls toward the sand. Snell's law, n₁ sin θ₁ = n₂ sin θ₂, is the precise bookkeeping of that swing, published by Willebrord Snellius in 1621 and derivable from Fermat's principle that light takes the fastest path.
Into a denser medium (n₂ > n₁), rays bend toward the normal; back into a thinner one, they bend away. That second case has a limit — and the limit is where the interesting optics live.
The critical angle and total internal reflection
Going from dense to thin, the refracted ray bends away from the normal, so at some incidence angle it would need to exit at more than 90° — impossible. Beyond that critical angle, θc = arcsin(n₂/n₁), the surface becomes a perfect mirror: total internal reflection. For glass-to-air it is about 41.8°, for water-to-air 48.6°. This is why fiber-optic cables carry light for kilometers with almost no loss, why diamonds are cut so facets keep light bouncing internally, and why the underwater view of the sky compresses into the 97°-wide Snell's window.
How it’s calculated
θ₂ = arcsin(n₁ sin θ₁ / n₂), with angles measured from the normal in degrees. When n₁ sin θ₁ / n₂ > 1 there is no real solution and the tool reports total internal reflection. Critical angle θc = arcsin(n₂/n₁), defined only when n₁ > n₂. Light speed in medium 2 = c/n₂ with c = 299,792,458 m/s exactly.
Indices are treated as single numbers, but real materials disperse — n varies a few percent across the visible band, which is exactly how prisms split white light.
Refractive indices of common materials (589 nm)
| Material | n | Light speed |
|---|---|---|
| Vacuum | 1 (exact) | 299,792 km/s |
| Air (sea level) | 1.0003 | 299,703 km/s |
| Water | 1.333 | 224,901 km/s |
| Ethanol | 1.361 | 220,274 km/s |
| Crown glass | 1.52 | 197,232 km/s |
| Sapphire | 1.77 | 169,374 km/s |
| Diamond | 2.417 | 124,035 km/s |
Standard optics-handbook values at the 589 nm sodium D line (e.g., CRC Handbook); speeds computed as c/n.
Common mistakes
- Measuring angles from the surface instead of from the normal — a ray skimming 10° above the glass is at 80° incidence, not 10°.
- Swapping n₁ and n₂; the bend direction reverses, and a possible refraction can turn into spurious total internal reflection.
- Expecting a critical angle when entering a denser medium — TIR only exists going from higher n to lower n.
- Using one index for all colors in dispersion-sensitive work; n for violet is measurably higher than for red in the same glass.
Frequently asked questions
What is Snell's law?
n₁ sin θ₁ = n₂ sin θ₂, where n is each medium's refractive index and the angles are measured from the normal. Solve for the exit angle: θ₂ = arcsin(n₁ sin θ₁ / n₂).
What happens when the calculator says total internal reflection?
Your incidence angle exceeds the critical angle arcsin(n₂/n₁), so sin θ₂ would have to be greater than 1 — no refracted ray can exist. All the light reflects back into the first medium.
How do I find a refractive index from measured angles?
Rearrange Snell's law: n₂ = n₁ sin θ₁ / sin θ₂. Shine a ray from air (n₁ ≈ 1.000) at a known angle, measure the refracted angle, and divide the sines — the classic lab method.
Is the angle measured from the surface or the normal?
Always from the normal, the line perpendicular to the boundary. This is the most common source of wrong answers — a ray hitting nearly flat along the surface has an incidence angle near 90°, not near 0°.
Why does a pool look shallower than it is?
Light from the bottom bends away from the normal as it exits into air, and your brain traces the rays straight back. The apparent depth is roughly the true depth divided by 1.333 — about 25% shallower.